Delta function of two variables

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this is about integration of delta function where both the arguments are variables

e.g $\delta(x-y)$ where both are x and y are variable now I am now trying to calculate the integral wrt x in following way,

$$\int f(x) \delta(x-y) dx = f(y)$$ where y is also variable. and $$\int f(y) \delta(x-y) dy = f(x)$$

Please let me know if i am doing it correctly

Thanks


Thanks for the clarification,, one more query can I write

$$\delta(x^2 -y^2) = \frac{\delta(x+y) + \delta(x-y)}{2y}$$ where x and y both are variables..

and

$$\delta(x-y) f(x) = f(y)$$

Please let me know..